CHARLES S. PEIRCE ON THE LOGIC OF NUMBER

Abstract

The topic of this thesis is a single brief paper written by Peirce in 1881, called "On the Logic of Number." Despite its brevity, Peirce's 1881 paper was one of the major achievements of the nineteenth century in the foundations of mathematics. It contained the first successful axiom system for the natural numbers. Since scholarship has traditionally attributed priority in this regard to the axiom systems of Richard Dedekind, in 1888, and Giuseppe Peano, in 1889, an important result of this thesis is its demonstration that Peirce's axiom system is actually equivalent to these latter.^ The technical sections of this thesis provide important historical background on the systems of Peirce, Dedekind, and Peano. It is pointed out that Peirce's 1881 paper gave the first abstract formulation of the notions of partial and linear order, and the first ordinal definition of cardinals. The approaches of all three to such topics as mathematical induction, use of recursive definitions, cardinality, categoricity, and the axiom of infinity are discussed. These sections culminate with the formal demonstration, in Chapter II, that Peirce's axiom system is equivalent to that of Dedekind. This proof employs onlyl a modest set theory with no special assumptions. It is more of historical than mathematical value.^ The significance of this proof goes beyond merely giving credit where credit is due. There is a widespread tendency to view the foundations of mathematics as having sprung full-grown from the heads of a few individuals, notably Frege, Peano, and Russell. The work of Peirce, and that of many other nineteenth century pioneers, from Boole to Schroder, is often neglected, or, at best, consigned to some vague "prel-history" of foundations. By establishing the priority of Peirce's axiom system, this thesis takes issue with the conventional wisdom and attempts to reinforce a more accurate perception of the gradual and continuous development of foundations in the nineteenth century.^ Peirce's interest in the foundations of mathematics was intimately related to his dominant philosophical concerns. Some of his most characteristic metaphysical and epistemological doctrines, e.g., synechism and the theory of the categories, bear the direct imprint of his work on sets and transfinite numbers. In particular, his 1881 paper is important for understanding Peirce's classification of the sciences. Although it has apparently escaped the attention of most Peirce scholarship, his 1881 paper was published simultaneously with his father's famous definition of mathematics as "the science which draws necessary conclusions." This historical connection is important because Peirce eventually adopted his father's definition as the centerpiece of his own philosophy of mathematics. Hence it indicates how Peirce himself understood the philosophical implications of his axiom system, and sheds light on how he viewed the relation between mathematics and logic, and thus the classification of the sciences as a whole.^ In its final section, this thesis addresses the problem of locating Peirce's mature philosophy of mathematics vis-a-vis the traditional positions of logicism, intuitionism, and formalism. Similarities and differences with all three are found, but the diffrences are emphasized. Perhaps the most important, and contemporary, single feature of Peirce's conception of mathematics is that he does not conceive it to require any foundation at all, whether in logic, in intuition, or by means of constructive completeness proofs. Mathematics, for Peirce, is essentially independent and self-sufficient. ^